Wage growth is breaking out in an unexpected corner of the U.S. economy: the nation’s restaurants and bars.

Food-service employment has surged since the recession ended nearly six years ago, growing twice as fast as overall payrolls. But those gains had largely failed to translate into better wages in the sector, until recently. Restaurant wages zoomed up to an annualized pace of more than 3% in the second half of last year from below a 1.5% pace in the first half of 2013, according to the Labor Department. Private-sector wages across the overall economy have grown at about a 2% pace for the past five years.

Many restaurant owners are now scrambling to hire and retain workers, a potential precursor to widespread wage gains if it signals diminished slack in the labor market.

Euclid Hospitality Group in St. Louis, operator of Pi Pizzerias, raised wages last year for kitchen staff and other employees who don’t earn tips in order to slow turnover and attract better candidates. Founder Chris Sommers said he is competing for workers, not just with restaurants, but with other industries.

Driving the brisker wage growth are a number of factors, including a higher minimum wage in many states, falling unemployment and stronger demand for meals outside the home, fueled by growing disposable incomes. Other sectors also may be feeling pressure. Big retailers such as Wal-Mart Stores Inc. and TJX Cos., the parent company of T.J. Maxx, for example, recently announced raises.

In past economic cycles, low-skilled workers have been among the last to see a pickup in wage growth. But the latest gains suggest a consumer-driven recovery could draw more Americans back into the job market and further bolster consumption, a key driver of economic expansion.

“After a few hard years, customers are treating themselves again,” said Jennifer Durham, vice president of franchise development at Checkers Drive-In Restaurants Inc. Higher sales led the burger chain to add 20 stores last year, hire hundreds of employees and raise its wages—with wage growth twice as fast in 2014 as it was in 2013.

Even part-time workers are in high demand, she said, a sign of wage pressure across the spectrum. “There is a lot of competition…That’s pressing us, in a positive way, to increase wages.”

Cheesecake Factory Inc. is now planning for wage inflation of about 3% after it registered just around 1% a year since the recession. “Wage-rate increases are a reality, not just for us but, if not yet, soon to be across the industry,” David Overton, chairman and chief executive, told investors in early February.

The deep recession left a glut of labor for restaurants to choose from—both higher-skilled workers who lost better paying jobs and the lower-skilled people the industry more typically hired. An excess supply of labor tends to depress wage growth. Now hiring has picked up in the broader economy, with overall payrolls growing at the best pace since the late 1990s in recent months. Skilled workers with restaurant jobs are frequently looking for work in other fields. As they leave kitchens and counters, it creates shortages of workers with food-service backgrounds.

“A lot of new [establishments] are popping up and those restaurants want experienced people,” said Patrice Rice, chief executive of Patrice & Associates, a restaurant staffing firm. “There is fierce competition.” Some of her clients are offering significant raises, paid vacation time and even free meals for spouses to attract workers. A national pizza franchise recently paid her $5,000 per position to place managers. Her fees are up 25% from a year ago.

The upshift in wages shows “there is a dearth of viable labor and employers are willing to pay up for that,” said Tom Porcelli, chief U.S. economist at RBC Capital Markets.

Outsized growth in restaurant wages might not be an entirely positive sign for the broader economy. In the previous two cycles, pay gains for restaurant workers peaked very late in the expansion, after broader wage growth plateaued. If escalating restaurant wages mark the end, rather than start, of wider wage growth, that would limit the economy’s ability to accelerate.

The industry is an engine of job creation—with payrolls growing by 16% since the recession ended in 2009—but its ability to influence the overall economy is limited because workers are paid less. Nonsupervisory food-service employees earned $11.11 an hour, on average, last year—compared with a national mean of $20.61 an hour. And about half of restaurant workers are part-timers.

Rising pay at restaurants “doesn’t necessarily translate into higher wages at call centers or in banking,” said Kevin Kliesen, economist at the Federal Reserve Bank of Kansas City. Rising wages for servers, dishwashers and cooks typically are a result of strong demand for labor in other fields, not a leading indicator signaling that pay will grow more broadly.

Stronger consumer demand, aided in part by falling gasoline prices, has fueled restaurant spending, which was up 11.3% over the past 12 months, more than in any other segment tracked by the government’s monthly retail and food-services report. For the first time on record, Americans are spending as much money dining out as at traditional grocery stores.

Food-service pay is also bolstered by 17 states increasing the minimum wage in 2014, including a 12.5% jump in California. Even more states raised pay at the start of this year. Nearly half of minimum-wage workers are employed in food service, according to the Labor Department.

Firehouse Subs stores in northeast Florida now pay an average of $8.50 an hour, up from $8.23 last year, partially reflecting a 12-cent bump in the state minimum wage to $8.05. In addition to better pay, Firehouse started offering health insurance to full-time employees last year and altered its scheduling system in an effort to keep turnover low and reward workers.

“There’s some ripple effect when minimum wage goes up,” said Chief Executive Don Fox, referring to salaries and benefits that employers must offer. “And I think it’s fair to say that as the economy has gradually improved, the labor market becomes a little bit more difficult.”

Write to Eric Morath at eric.morath@wsj.com and Jeffrey Sparshott at jeffrey.sparshott@wsj.com